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Spacetime Is a Manifold That Is Continuous and Differentiable. This I. GENERAL RELATIVITY { A SUMMARY A. Pseudo-Riemannian manifolds Spacetime is a manifold that is continuous and differentiable. This means that we can define scalars, vectors, 1-forms and in general tensor fields and are able to take derivatives at any point. A differential manifold is an primitive amorphous collection of points (events in the case of spacetime). Locally, these points are ordered as points in a Euclidian space. Next, we specify a distance concept by adding a metric g, which contains information about how fast clocks proceed and what are the distances between points. −−! On the surface of the Earth we can determine a metric by drawing small vectors ∆P on the surface. We state that the length of the vector is given by the inner product −−! −−! −−! −−! ∆P· ∆P ≡ ∆P2 = (length of ∆P)2; (1.1) and use a ruler to determine its value. We now have a definition for the inner vector product for a small vector with itself. We use linearity to extend this to macroscopic vectors. Next, we can obtain a definition for the inner product of two different vectors by writing 1 h i A~ · B~ = (A~ + B~ )2 − (A~ − B~ )2 : (1.2) 4 In summary, when one has a distance concept (a ruler on the surface of the Earth), then one can define an inner product, and from this the metric follows (since it is nothing but g(A;~ B~ ) ≡ (A~ · B~ ) = g(B;~ A~). The metric tensor is symmetric.). A differentiable manifold with a metric as additional structure, is termed a (pseudo-)Riemannian manifold. We now Figure 1: Left: at each point P on the surface of the Earth a tangent space (in this case a tangent plane) exists; right: the tangent plane is a nearly correct image in the vicinity of the point P. want to assign a metric to spacetime. To this end we introduce a local Lorentz frame (LLF). We can achieve this by going into freefall at point P. The equivalence principle states that all effects of gravitation disappear and that we locally obtain the metric of the special theory 1 of relativity (SRT). This is the Minkowski metric. Thus, we can choose at each point P of the manifold a coordinate system in which the Minkowski metric is valid. While in the SRT this can be a global coordinate system, in general relativity (GR) this is only locally possible. With this procedure we have now found a definition of distance at each point P: 2 µ ν with gµν = ηµν ! ds = ηµνdx dx . In essence, we practice SRT at each point P and have a measure for lengths of rods and proper times of ideal clocks. In a LLF the metric is given by ηµν = diag(−1; +1; +1; +1). For a Riemannian manifold all diagonal elements need to be positive. The signature (the sum of the diagonal elements) of the metric of spacetime is +2, and in our case we refer to the manifold as pseudo-Riemannian. Assume that we draw a coordinate system on the Earth's surface with longitude and latitude. When we look at this reference system, it locally resembles a Cartesian system, when we stay close to point P. Deviations from Cartesian coordinates occur at second order in the distance x from the point P. Mathematically, this means that j~xj2 g = δ + O ; (1.3) jk jk R2 with R the radius of the Earth. A simpler way to understand this is by constructing the tangent plane at point P. Fig. 1 shows that when ~x denotes the position vector of a point with respect to P, then this corresponds to cos j~xj on the tangent plane. A series expansion x2 yields cos x = 1 − 2 + :::. As a consequence we see that when one considers only first-order derivates, one observes no influence of the curvature of the Earth. Only when second-order derivatives are taken into account, one obeys curvature effects. The same is true for spacetime. In a curved spacetime we cannot define a global Lorentz frame for which gαβ = ηαβ. However, it is possible to choose coordinates such that in the vicinity of P this equation is almost valid. This is made possible by the equivalence principle. This is the exact definition of a local Lorentz frame and for such a coordinate system one has gαβ(P) = ηαβ for all α; β; @ @xγ gαβ(P) = 0 for all α; β; γ; (1.4) @2 @xγ @xµ gαβ(P) 6= 0: The existence of local Lorentz frames expresses that each curved spacetime has at each point a flat tangent space. All tensor manipulations occur in this tangent space. The above expressions constitute the mathematical definition of the fact that the equivalence principle allows us to chose a LLF at point P. The metric is used to define the length of a curve. When d~x is a small vector displacement 2 α β on a curve, then the quadratic length is equal to ds = gαβdx dx (we call this the line element). A measure for the length is found by taking the root of the absolute value. This α β 1 yields dl ≡ jgαβdx dx j 2 . Integration gives the total length l and we find 1 Z 1 Z λ1 α β 2 α β 2 dx dx l = gαβdx dx = gαβ dλ, (1.5) along the curve λ0 dλ dλ 2 where λ is the parameter of the curve. The curve has as end points λ0 and λ1. The tangent vector V~ of the curve has components V α = dxα/dλ and we obtain Z λ1 1 ~ ~ 2 l = V · V dλ (1.6) λ0 for the length of an arbitrary curve. When we perform integrations in spacetime it is important to calculate volumes. With volume we mean a four-dimensional volume. Suppose that we are in a LLF and have 0 1 2 3 α a volume element dx dx dx dx , with coordinates fx g in the local Lorentz metric ηαβ. Transformation theory states that 0 1 2 3 @(x ; x ; x ; x ) 0 0 0 0 dx0dx1dx2dx3 = dx0 dx1 dx2 dx3 ; (1.7) @(x00 ; x10 ; x20 ; x30 ) where the factor @()=@( ) is the Jacobian of the transformation of fxα0 g to fxαg. One has 0 @x0 @x0 1 0 0 ::: @(x0; x1; x2; x3) @x0 @x1 @x1 @x1 α 0 0 0 0 = det @ 0 0 ::: A = det Λ β0 : (1.8) @(x0 ; x1 ; x2 ; x3 ) @x0 @x1 ::: ::: ::: The calculation of this determinant is rather evolved and it is simpler to realize that in terms of matrices the transformation of the components of the metric is given by the equation (g) = (Λ)(η)(Λ)T , where with `T ' the transpose is implied. Then the determinants obey det(g) = det(Λ)det(η)det(ΛT ). For each matrix one has det(Λ) = det(ΛT ) and furthermore we have det(η) = −1. We obtain det(g) = − [det(Λ)]2. We use the notation 1 α 2 g ≡ det(gα0β0 ) ! det(Λ β0 ) = (−g) (1.9) and find 0 1 2 3 1 00 10 20 30 1 00 10 20 30 dx dx dx dx = det [−(gα0β0 )] 2 dx dx dx dx = (−g) 2 dx dx dx dx : (1.10) It is important to appreciate the reasoning we followed in order to obtain the above result. We started in a special coordinate system, the LLF, where the Minkowski metric is valid. We then generalized the result to all coordinate systems. B. Tensors and covariant derivative Suppose we have a tensor field T( ; ; ) with rank 3. This field is a function of location and defines a tensor at each point P. We can expand this tensor in the basis f~eαg which gives the (upper-index) components T αβγ. In general we have 64 components for spacetime. However, we also can expand the tensor T in the dual basis f~e αg and we find αβγ γ α β T( ; ; ) ≡ T ~eα ⊗ ~eβ ⊗ ~eγ = Tαβ ~e ⊗ ~e ⊗ ~eγ: (1.11) When we want to calculate the components we use the following theorem: αβγ α β γ γ γ T = T(~e ;~e ;~e ) and Tµν = T(~eµ;~eν;~e ): (1.12) 3 When we have the components of tensor T in a certain order of upper and lower indices, and we want to know the components with some other order of indices, then the metric can be used. One has γ αβγ αβγ αρ βγ Tµν = T gαµgβν and for example also T = g Tρ (1.13) Next, we want to discuss contraction. This is rather complicated to treat in our abstract notation. Given a tensor R, we always can write it in terms of a vector basis as R( ; ; ; ) = A~ ⊗ B~ ⊗ C~ ⊗ D~ + ::: (1.14) We discuss contraction only for a tensor product of vectors and use linearity to obtain a mathematical description for arbitrary tensors. For contraction C13 of the first and third index one has h ~ ~ ~ ~ i ~ ~ ~ ~ C13 A ⊗ B ⊗ C ⊗ D( ; ; ; ) ≡ (A · C)B ⊗ D( ; ): (1.15) We can write the above abstract definition in terms of components and find ~ ~ µ ν µ ν µ µβ δ ~ ~ A · C = A C ~eµ · ~eν = A C gµν = A Cµ ! C13R = R µ B × D: (1.16) In the same way as above, we see that from two vectors A~ and B~ a tensor A~ ⊗ B~ can be constructed by taking the tensor product, while we can obtain a scalar A~ · B~ by taking the inner product. The contraction of the tensor product A~ ⊗ B~ again yields a scalar, h i C A~ ⊗ B~ = A~ · B~ .
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